The fact that there are two covenants, one old and another new points to the fact that one is fulfilled, or "abolished." Once you pay your mortgage on a house, I would recommend you stop sending payments to the bank. Its been fulfilled, in the same way we are no longer under Law trying to fulfill its requirements. Jesus did that on our behalf, and has imputed His righteousness to us.
I don't see why it sickens you when people say a particular part of God's Word doesn't apply to us, as we have gone from one covenant to another. You can't take that which is of the old and mix it with the new, unless the new states it as well. Its like a lot of the things Jesus said before He died on the cross and resurrected on the third day, it was to people under the Law, not in the new covenant reality of grace. For example, forgive others or the Lord won't forgive you. Thats an old testament principle. In the new it is stated that we forgive others as the Lord has forgiven us, meaning we already have the forgiveness. That is just one example.
Its all about rightly dividing the Word. Don't put Law into grace, because you then nullify it. That is why there is a distinction, that is why somethings are now 'void.' Mixture or balancing of Law and grace takes away from both, one is self-righteousness and the other imputed, no work of your own. You can't mix the two, it doesn't work.
All of Jesus' teachings are found in and founded upon the Old Covenants; but not all of the duties and requirements of the Old Covenants are included in the New Covenant.
The Levitical Priesthood and the ceremonial requirements of the LAW are superceded by the high priesthood of Jesus, and the priesthood of the believer.
All of the moral and ethical precepts of the OT are restated more forcefully in the NT and the New Covenant of Jeremiah 31:31-34 as amplified in the teachings of Jesus retains them.
The Old Covenants (there were several) all depended on the actions and observances of the people under them.
The New Covenant,
Jer 31:31-34
31 Behold, the days come, saith the Lord, that I will make a new covenant with the house of Israel, and with the house of Judah:
32 Not according to the covenant that I made with their fathers in the day that I took them by the hand to bring them out of the land of Egypt; which my covenant they brake, although I was an husband unto them, saith the Lord:
33 But this shall be the covenant that I will make with the house of Israel; After those days, saith the Lord, I will put my law in their inward parts, and write it in their hearts; and will be their God, and they shall be my people.
34 And they shall teach no more every man his neighbour, and every man his brother, saying, Know the Lord: for they shall all know me, from the least of them unto the greatest of them, saith the Lord; for I will forgive their iniquity, and I will remember their sin no more.
KJV
depends upon ONLY what God has done.
Under the New Covenant, the work of God's people consists of what God does in us; and NOT what we attempt to do for Him.